[Paper Review] Generalized Dantzig Selector: Application to the k-support norm

Soumyadeep Chatterjee, Sheng Chen+1 more|arXiv (Cornell University)|Jun 20, 2014

Sparse and Compressive Sensing Techniques21 references17 citations

TL;DR

This paper introduces the Generalized Dantzig Selector (GDS), a flexible framework for structured estimation in linear models using any norm, with a focus on the k-support norm. It proposes an inexact ADMM algorithm leveraging conjugate proximal operators and establishes non-asymptotic high-probability bounds on estimation error using Gaussian width analysis, providing the first statistical recovery guarantee for the k-support norm with explicit computational and theoretical convergence rates.

ABSTRACT

We propose a Generalized Dantzig Selector (GDS) for linear models, in which any norm encoding the parameter structure can be leveraged for estimation. We investigate both computational and statistical aspects of the GDS. Based on conjugate proximal operator, a flexible inexact ADMM framework is designed for solving GDS, and non-asymptotic high-probability bounds are established on the estimation error, which rely on Gaussian width of unit norm ball and suitable set encompassing estimation error. Further, we consider a non-trivial example of the GDS using $k$-support norm. We derive an efficient method to compute the proximal operator for $k$-support norm since existing methods are inapplicable in this setting. For statistical analysis, we provide upper bounds for the Gaussian widths needed in the GDS analysis, yielding the first statistical recovery guarantee for estimation with the $k$-support norm. The experimental results confirm our theoretical analysis.

Motivation & Objective

• To develop a general framework for structured estimation in linear models using arbitrary norms, extending the classical Dantzig Selector beyond the L1 norm.
• To address the lack of computational and statistical theory for non-L1 norms in Dantzig-type estimators, particularly for the k-support norm.
• To design an efficient inexact ADMM algorithm that leverages proximal operators of the norm and its conjugate for scalable optimization.
• To establish non-asymptotic high-probability bounds on estimation error that depend on the Gaussian width of the norm's unit ball and error set.
• To provide the first statistical recovery guarantee for the k-support norm by deriving upper bounds on relevant Gaussian widths.

Proposed method

• Proposes the Generalized Dantzig Selector (GDS) as a convex optimization problem minimizing a norm R(θ) subject to a dual norm constraint on the residual gradient, R*(X^T(y - Xθ)) ≤ λp.
• Develops an inexact ADMM framework where primal updates correspond to proximal operators of R(θ) and its conjugate, enabling efficient computation via Moreau decomposition.
• Derives an efficient algorithm to compute the proximal operator of the k-support norm in O(p log p + log k log(p - k)) time, overcoming limitations of existing methods that only handle the squared norm.
• Uses Gaussian width analysis to bound estimation error, relying on the width of the unit norm ball and the error set, with theoretical bounds derived via concentration inequalities and group-wise Gaussian maxima.
• Applies Lemma 3 to bound the expected maximum ℓ2 norm over k-sized groups of Gaussian variables, enabling upper bounds on Gaussian widths for the k-support norm.
• Establishes high-probability bounds on estimation error by relating the dual norm of the residual to the Gaussian width of the error set, ensuring statistical consistency.

Experimental results

Research questions

• RQ1 Can the Dantzig Selector be generalized to arbitrary norms beyond L1, enabling structured estimation for complex parameter structures?
• RQ2 How can an efficient and scalable optimization algorithm be designed for the generalized Dantzig Selector with arbitrary norms?
• RQ3 What is the statistical performance of the GDS, and can non-asymptotic high-probability error bounds be established using geometric properties like Gaussian width?
• RQ4 What are the Gaussian width bounds for the k-support norm's unit ball and the error set, and how do they affect estimation error in high-dimensional settings?
• RQ5 Can the proximal operator for the k-support norm be computed efficiently, and does this enable practical implementation of the GDS in high-dimensional sparse learning?

Key findings

• The GDS framework enables structured estimation using any norm, with the dual norm constraint ensuring robustness to noise and adaptivity to parameter structure.
• An inexact ADMM algorithm is proposed that computes proximal updates for both the norm and its conjugate, enabling scalable optimization via Moreau decomposition.
• The k-support norm's proximal operator is computed efficiently in O(p log p + log k log(p - k)) time, a novel contribution as prior methods only applied to the squared norm.
• The Gaussian width of the k-support unit ball is upper bounded by O(√(k log(pe/k)) + √k), which is critical for deriving estimation error bounds.
• The estimation error of the GDS with k-support norm is bounded with high probability by O(√(k log(pe/k)) + √k), establishing the first statistical recovery guarantee for this norm.
• Experimental results confirm the theoretical analysis, showing that the GDS with k-support norm achieves strong performance in high-dimensional sparse estimation tasks.